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Unsound ordinals

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An ordinal zeta is unsound if there are subsets A_n (n in omega) of it such that as b ranges through the subsets of omega, uncountably many ordertypes are realised by the sets $\bigcup_{n \in b} A_n$.

Woodin in 1982 raised the question whether unsound ordinals ordinals exist; the answer I found then (to be found in a paper published in 1984 in Math Proc Cam Phil Soc) is this:

Assume DC. Then the following are equivalent:

i) the ordinal $\omega_1^{\omega + 2}$ (ordinal exponentiation) is unsound

ii) there is an uncountable well-ordered set of reals

That implies that if omega_1 is regular and the ordinal mentioned in i) is sound, then omega_1 is strongly inaccessible in the constructible universe. Under DC, every ordinal strictly less than the ordinal mentioned in i) is sound.

There are many open questions in this area: in particular, in Solovay’s famous model where all sets of reals are Lebesgue measurable, is every ordinal sound ? The question may be delicate, as Kechris and Woodin have shown that if the Axiom of Determinacy is true then there is an unsound ordinal less than omega_2.

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